An Infinite Class of Balanced Functions with Optimal Algebraic Immunity, Good Immunity to Fast Algebraic Attacks and Good Nonlinearity

Abstract

Abstract. After the improvement by Courtois and Meier of the algebraic attacks on stream ciphers and the introduction of the related notion of algebraic immunity, several constructions of infinite classes of Boolean functions with optimum algebraic immunity have been proposed. All of them gave functions whose algebraic degrees are high enough for resisting the Berlekamp-Massey attack and the recent Rønjom-Helleseth attack, but whose nonlinearities either achieve the worst possible value (given by Lobanov's bound) or are slightly superior to it. Hence, these functions do not allow resistance to fast correlation attacks. Moreover, they do not behave well with respect to fast algebraic attacks. In this paper, we study an infinite class of functions which achieve an optimum algebraic immunity. We prove that they have an optimum algebraic degree and a much better nonlinearity than all the previously obtained infinite classes of functions. We check that, at least for small values of the number of variables, the functions of this class have in fact a very good nonlinearity and also a good behavior against fast algebraic attacks.

On a Conjecture about Binary Strings Distribution

“…They also proved that, if the conjecture is true, then one can get in even dimension balanced Boolean functions of optimal algebraic immunity and of high nonlinearity (better than that of the function proposed in [3]). The approach of the authors was to identify annihilators of the Boolean functions in n variables that they consider with codewords of BCH codes.…”

“…In 2008, Carlet and Feng [3] proposed for the first time an infinite class of functions which seems able to satisfy all of the main criteria for being used as a filtering function in a stream cipher. Their functions are balanced with optimal algebraic degree, optimal algebraic immunity, good immunity to Fast Algebraic Attacks and good nonlinearity.…”

A Proof of Constructions for Balanced Boolean Function with Optimum Algebraic Immunity

“…But the computational complexity of the construction do not have been well studied. Carlet and Feng [11] proposed a well construction based on the Boolean functions' trace representation. Their Boolean functions have not only optimum AI but also high nonlinearity.…”

Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

“…f has maximum degree n − 1 and algebraic immunity n/2 . Besides, it reaches a high nonlinearity, which is better than [10]. Now, a class of 1-resilient Boolean functions which are still degree maximized and algebraic immunity optimized has been obtained by using f aforementioned.…”

“…The symmetry of the functions presents a risk if attacks using this peculiarity can be found in the future. Recently, when the number of variables n only equals 6,8,10,12, [2] provided 1-resilient functions with maximum degree and optimal algebraic immunity by a primary construction. Bars and Viola in [3] are trying to find a complete combinatorial characterization of well-behaved functions and good random generation algorithms for well-behaved functions.…”